• ELF.SEQ.E Exponential Equations

ELF.SEQ.E Exponential Equations

Consider the function $f(x) = Ab^{kx}$.

A standard form of an exponential equation with variable $x$ is an equation of the form $f(x) = Ab^{kx} = C$ with $A, k \ne 0$.
Intermediate algebra spends a considerable amount of time solving this equation without any reference to functions.
When $k = 1$ this equation has the solution $x = \log_b( \frac C A)$.
When $k \ne 1$ the equation can be solved so that $x = \frac 1k \log_b( \frac C A)$.

Example  SEQ.E.1 : Suppose $5*2^x= 20$. Find $x$.

Example SEQ.E.2 :Suppose $5*2^{4x}= 20$. Find $x$.
A slightly more ambitious form of an exponential equation with variable $x$ is an equation of the form $A_1b^{k_1x} = A_2b^{k_2x}$ with $A_1 \ne A_2$ and $k_1 \ne k_2$. Algebraically this is solved by solving the related equation $Ab^{kx} = 1$ where $A =\frac {A_1}{A_2}$ and $k = k_1 - k_2$.

Example SEQ.E.3 : Suppose $3* 2^{x-1} +1 = 13$. Find $x$. This example shows an important connection between a mapping diagram for an exponential function as a composition with core linear functions and the algebra used in solving an exponential equation. The included GeoGebra mapping diagram can be used to  visualize the algebra for solving equations of the form $A* b^{x-h} +k = C$
You can use this next dynamic example to solve exponential equations like those in Examples SEQ.E.1 and SEQ.E.2 visually with a mapping diagram of $f$ and the lines in the graph of $f$.
Example ELF.DSEQ.E.0 Dynamic Views for solving an equation $f(x) = Ab^{kx} = C$ on Graphs and Mapping Diagrams

ELF.SEQ.L Logarithmic Equations

Consider the function $f(x) = A\log_b(kx)$.

A standard form of a logarithmic equation with variable $x$ is an equation of the form $f(x) = A\log_b(kx)= C$ with $A, k \ne 0$.
Intermediate algebra spends a considerable amount of time solving this equation without any reference to functions.
When $k = 1$ this equation has the solution  $x = b^{ \frac C A}$.
When $k \ne 1$ the equation can be solved so that $x = \frac 1k b^{ \frac C A}$.

Example  SEQ.L.1 : Suppose $5*\log_2(x) = 20$. Find $x$.

Example SEQ.L.2 :Suppose $5*\log_2(4x) = 20$. Find $x$.

Example SEQ.L.3 : Suppose $3* \log_2(x+1) +1 = 13$. Find $x$. This example shows an important connection between a mapping diagram for a logarithmic function as a composition with core linear functions and the algebra used in solving a logarithmic equation. The included Geogebra mapping diagram can be used visualize the algebra for solving equations of the form $A* \log_b(x-h) +k = C$
You can use this next dynamic example to solve logarithmic equations like those in Examples SEQ.I.1 and SEQ.L.2 visually with a mapping diagram of $f$ and the lines in the graph of $f$.

Example ELF.DSEQ.L.0 Dynamic Views for solving an equation $f(x) = A\log_b(kx) = C$ on Graphs and Mapping Diagrams